Through trial and error, I can tell you that it doesn't serve much of a purpose to own two identical, or virtually identical guitars. There is a wide, wonderful world of tone out there that different guitars can give you. Investing in two of the same guitars is a little like a painter having two identical brushes.
I do understand the appeal of the LP. There's a classic vibe to it that can't be gained (for better or for worse) with other designs. I will say that the LP really isn't what it used to be unless you buy a Custom Shop reissue and those are big $$$. As it is, an LP standard will run you much more than a smokin' EBMM. Also keep in mind that standard LP's have weight relief holes to make the wood lighter (the custom shop models don't because they use better grades of wood). The latest models are actually chambered to make them lighter. Gibson says this is to make them better but I think it's so they can use even cheaper wood. Gibson products, on average, are way over priced based on the quality you can get from other companies such as EBMM. You pay a lot for the name.
All that aside, What vibe are you looking for? Vintage tone? An LP with vintage style PAF's would be a good choice. I don't really care a lot for Gibson pickups but the Burstbuckers are pretty good for this. A large variety of aftermarket PAF's are available so you can customize your tone. I love the Axis/EVH pickups - some of my very favorites - but they are not "vintage".
If you are interested in a good LP with great qc, for not a lot of $$$ do some research on the Epiphone Elitists. They are made in Japan, and in general, are made to Gibson C.S. spec. They use African Mahogany with no weight relief, Gibson hardware and pickups, long neck tenon, bone nut and sell new for around $1,200. They do use poly finish instead of nitro which is a huge deal to people in Gibson land. Not so much to people on the outside. I own one and it is the real deal for a fraction of the price. You do have to live without "Gibson" on the headstock though which is a shot to some people's fragile egos...
I'm giving you the above because info on EBMM products is easy to find here. Your other option is to buy another EBMM guitar that isn't an Axis. Honestly, I don't know if EBMM puts out a guitar with a "vintage" type tone but if they do, that's certainly an option. QC is certainly more of a sure thing with EBMM.
If you go the Gibson route, make sure you play the guitar before you buy. Their qc isn't the greatest and their products swing pretty wildly in quality and playability.
Hope this helps. Good luck and have fun!